Another reason why homotopy groups are infuriating

I’ve dabbled in higher homotopy groups before mostly to define things like higher K-theory but have never spent much time getting to know them. I was intrigued when I heard that \pi_n(X) is naturally an integral representation of \pi_1(X) via the standard monodromy action. This means that, if you tensor with some field of characteristic zero, you end up with a ‘proper’ representation of the fundamental group, or, equivalently, a local system on X. These objects arose quite naturally, so maybe they’re interesting…

I know of no interesting examples. First, higher homotopy groups are often torsion so the tensored representation is frequently going to be zero. Second, in many cases where there are nice descriptions of things like \pi_n(X)\otimes\mathbb{C} the monodromy action is trivial. This is the case for Lie groups, generalising the fact that \pi_1(G) is abelian and so the monodromy = conjugation action is trivial.

Serre proved in the 1950s that the action of \pi_1(X) is trivial when X is an H-space: a topological unital magma (yep, I just wanted to use the word ‘magma’). Here’s my slightly sorrowful note proving Serre’s result (from which one can deduce that \pi_1(X) is abelian for X an H-space and so, in particular, the case for Lie groups).

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